Theorem xrsle

Description: The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015)

	Ref	Expression
Assertion	xrsle	$$⊢\le ={\le }_{{ℝ}_{𝑠}^{*}}$$

Proof

Step	Hyp	Ref	Expression
1		xrex	$$⊢{ℝ}^{*}\in \mathrm{V}$$
2	1 1	xpex	$$⊢{ℝ}^{*}\times {ℝ}^{*}\in \mathrm{V}$$
3		lerelxr	$$⊢\le \subseteq {ℝ}^{*}\times {ℝ}^{*}$$
4	2 3	ssexi	$$⊢\le \in \mathrm{V}$$
5		df-xrs	$$⊢{ℝ}_{𝑠}^{*}=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},{ℝ}^{*}⟩,⟨{+}_{\mathrm{ndx}},{+}_{𝑒}⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_{𝑒}⟩\right\}\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{ordTop}\left(\le \right)⟩,⟨{\le }_{\mathrm{ndx}},\le ⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(x\in {ℝ}^{*},y\in {ℝ}^{*}⟼if\left(x\le y,y{+}_{𝑒}\left(-x\right),x{+}_{𝑒}\left(-y\right)\right)\right)⟩\right\}$$
6	5	odrngle	$$⊢\le \in \mathrm{V}\to \le ={\le }_{{ℝ}_{𝑠}^{*}}$$
7	4 6	ax-mp	$$⊢\le ={\le }_{{ℝ}_{𝑠}^{*}}$$